Linear mixed integer programming g p g gfor topology optimization of 

trusses and plates

Makoto Ohsaki and Ryo Watada

Kyoto University, JAPAN

Michell trussMichell truss

A.G.M. Michell, The limits of economy inframe structures 1904

Infinite number of members in

frame structures, 1904.

W Prager A note on discretizedprincipal stress line

W. Prager, A note on discretized Michell structure,1974

Finite number of membersConsider nodal cost

Ground structure approach to topology optimization

W. Dorn et al., Automatic design of optimal structures, 1964

1 A i ll th d d b1. Assign all the nodes and members.

2. Optimize cross‐sectional areas as continuous variables.

3. Remove unnecessary members and nodes.y

Optimization under stress constraints

Single loading condition

Statically determinateMultiple loading condition Statically determinate fully stressed

Statically indeterminate ygenerally not fully stressed

Formulation of topologyFormulation of topology optimization under stress constraintsoptimization under stress constraints

Minimize total structural volume

UkL σσσ ≤≤ )(A 0≥Afs. t. iii σσσ ≤≤ )(A 0≥iAfor

Stress constraints are given only forStress constraints are given only for existing members

iA : cross-sectional area of member i

)(Akσ : stress of member i for load k)(Aiσ : stress of member i for load k

Multiple loading conditionsMultiple loading conditions

Optimal topology

Member 3 does not existMax. stress ratio = 4.03

V = 12.812

Fully stressed design Optimal topology under multiple loading conditions cannot beloading conditions cannot be obtained by conventionalground structure approach

V = 15.986

Difficulties in topology optimization of trusses

• No stress constraint for non‐existing member.

• Discontinuity in problem formulation.St t i t dd l di– Stress constraint suddenly disappear as area approach 0.

• Optimal solution may exist at a singular  Feasible regiongpoint (cusp).

Optimal solution

Unstable optimal solutionUnstable optimal solution (member buckling)(member buckling)

i ioptimize

Fix node 2 different slenderness ratioFix node 2 different slenderness ratiodifferent stress bound

Infeasible optimal solutionsInfeasible optimal solutions

Intersection of members

Very thin member

Unstable node

d fMixed integer programming for truss topology optimizationtruss topology optimization

with discrete variableswith discrete variables

Topology optimization problemTopology optimization problem

minimize V a l= ∑ Structural volumeminimize

subject to

j jj

V a l=

=

∑Bs f

Structural volume

equilibrium

min max

subject to j j j j ja s a

Eσ σ≤ ≤

Bs fStress constraint

q

j Tj j j

j

Es a

l= b uForce constraint Nonlinear relation

0j

ja ≥V i bl

Select from list of available sectionsjaVariables: s, u, ja

Select from available listSelect from available list{0 }a a a∈ =A 1{0, , , }

jj j j jNa a a∈ =A

⎧0-1variable

1 if 0 otherwise

j jkjk

a ax

=⎧= ⎨⎩0 otherwise⎩

Express using jkxN N

1 1

, 1j jN N

j jk jk jk jk k

a x a x x= =

= = ≤∑ ∑　　

1,

jNj T

jk jk jk j j jkk

Es x a s s

l= =∑b u

1kjl =Stress-displ. relation

Linearization of stress disp relationLinearization of stress‐disp. relation

Linearization of nonlinear constraints(Stolpe and Svanberg)( p g)

0 1TiEs x a x orb umin max

ik ik ikx c s x c≤ ≤

, 0 1iik ik ik i ik

i

s x a x orl

= =b u

min T max(1 ) (1 )

ik ik ik

ik ik j ik iki

Ex c a s x cl

− ≤ − ≤ −b uil

Cmax, Cmin: sufficiently large/small value

Mixed integer linear programming (MILP)

Optimization of bridge truss from traditional configurations

SelectionTraditional configurations

Select from three types

Howe truss Platt truss

⇒ combination of traditionalconfigurations.Warren truss K‐truss

Optimization results by CPLEXOptimization results by CPLEXCase 1: 40.7 10 V=65.410 CPU = 67900(sec)f −= ×

4Case 2: 41.0 10 V=74.610 CPU = 14600(sec)f −= ×

Case 3: 41.4 10 V=81.203 CPU = 10600(sec)f −= ×

Warren truss for small loads.

⇒ Howe truss near support for large loads. ( {0.0,1.0,1.8})=A

CPU time strongly depends on load magnitude.

Optimization of space trussOptimization of space truss

Loaded node

Optimization of space truss• Ground structure Selection pattern

Traditional configurationsTraditional configurations

A: Schuwedler domeB: Lamella domeB: Lamella dome

Schuwedler dome Lamella dome

Optimization resultsOptimization resultsCase 1: 41.0 10 V=1404.3f −= × Case 2: 43.0 10 V=1441.1f −= ×

4 4Case 3: 41.0 10 V=1404.3f −= × Case 4: 410.0 10 V=1527.2f −= ×

Lamella dome for small loads.

( {0.0,1.0,2.0})=A

⇒ Schuwedler dome in perimeter region for large loads.